**Three-Dimensional Hydro-Magnetic Flow Arising in a Long Porous Slider and a Circular Porous Slider with Velocity Slip**

### **Naeem Faraz 1,\*, Yasir Khan 2, Amna Anjum 1,3,\* and Muhammad Kahshan 4,5**


Received: 21 July 2019; Accepted: 12 August 2019; Published: 16 August 2019

**Abstract:** The current research explores the injection of a viscous fluid through a moving flat plate with a transverse uniform magneto-hydrodynamic (MHD) flow field to reduce sliding drag. Two cases of velocity slip between the slider and the ground are studied: a long slider and a circular slider. Solving the porous slider problem is applicable to fluid-cushioned porous sliders, which are useful in reducing the frictional resistance of moving bodies. By using a similarity transformation, three dimensional Navier–Stokes equations are converted into coupled nonlinear ordinary differential equations. The resulting nonlinear boundary value problem was solved analytically using the homotopy analysis method (HAM). The HAM provided a fast convergen<sup>t</sup> series solution, showing that this method is efficient, accurate, and has many advantages over the other existing methods. Solutions were obtained for the different values of Reynolds numbers (R), velocity slip, and magnetic fields. It was found that surface slip and Reynolds number had substantial influence on the lift and drag of the long and the circular sliders. Moreover, the effects of the applied magnetic field on the velocity components, load-carrying capacity, and friction force are discussed in detail with the aid of graphs and tables.

**Keywords:** porous slider; MHD flow; reynolds number; velocity slip; homotopy analysis method
